Non-linear integer programming fleet assignment model

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NON-LINEAR INTEGER PROGRAMMING

FLEET ASSIGNMENT MODEL

Prince Lerato Phokomela

A dissertation submitted to the Faculty of Engineering and

the Built Environment, University of the Witwatersrand,

Johannesburg, in fulfilment of the requirements for the

degree of Master of Science in Engineering.

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Declaration

I declare that this dissertation is my own unaided work. It is submitted for the degree of Master of Science to the University of the Witwatersrand, Jo-hannesburg. It has not been submitted before for any degree or examination to any other University.

... (Signature of Candidate)

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Abstract

Given a flight schedule with fixed departure times and cost, solving the fleet assignment problem assists airlines to find the minimum cost or maximum revenue assignment of aircraft types to flights. The result is that each flight is covered exactly once by an aircraft and the assignment can be flown using the available number of aircraft of each fleet type.

This research proposes a novel, non-linear integer programming fleet assign-ment model which differs from the linear time-space multi-commodity network fleet assignment model which is commonly used in industry. The performance of the proposed model with respect to the amount of time it takes to create a flight schedule is measured. Similarly, the performance of the time-space multi-commodity fleet assignment model is also measured. The objective function from both mathematical models is then compared and results reported.

Due to the non-linearity of the proposed model, a genetic algorithm (GA) is used to find a solution. The time taken by the GA is slow. The objective function value, however, is the same as that obtained using the time-space multi-commodity network flow model.

The proposed mathematical model has advantages in that the solution is easier to interpret. It also simultaneously solves fleet assignment as well as individual aircraft routing. The result may therefore aid in integrating more airline planning decisions such as maintenance routing.

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In memory of my mother

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Acknowledgements

My deepest gratitude goes to:

My supervisors, Dr Ian Campbell and Prof. M. Ali for agreeing to assist me produce this body of work. Without their depth of knowledge and constant

motivation, I would not have been able to finish such an arduous task of producing this document.

Fujitsu for inspiring the problem and providing the data for testing.

My wife, Akhona Phokomela for her patience, understanding and for consistently spurring me on during this endeavour.

My brother, Mandla Mashaba and the rest of my family for the constant support.

My friend, Sefako Tholo for all the technical assistance.

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the aircraft-time line 166 J Data set 1 solution results 167 K Data set 2 solution results 168 L Data set 3 solution results 169 M Data set 4 solution results 170 N Data set 5 solution results 171 O Data set 6 solution results 172 P Data set 7 solution results 173 Q Data set 8 solution results 174 R Data set 9 solution results 175

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S GA solver compliance with all NLIP FAM constraints 176

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List of Figures

Page

2.1 The number of active point-to-point air services by year (Pearce

2013) . . . 21

2.2 An example of a typical hub-and-spoke network (Flynn 2016) . 22 2.3 Overview of airline planning process (Lohatepanont 2002) . . . . 24

2.4 The Domestic Airline Scheduling Process at a major US airline (Goodstein 1997) . . . 28

3.1 Research published for airline scheduling decisions . . . 42

3.2 A representation of a time-space multi-commodity network flow fleet assignment model (MCNF FAM) . . . 43

4.1 A graphical depiction of the proposed fleet assignment model with parameter descriptors . . . 49

4.2 Flights sorted by departure time . . . 55

5.1 An example of a chromosome with no flights assigned . . . 66

5.2 Example of mutation operator . . . 71

5.3 Example of the single point crossover operator . . . 72

5.4 Example of the double point crossover operator . . . 73

5.5 Example of the uniform crossover operator . . . 74

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6.1 Process flow for the non-linear integer programming fleet

assignment model using GA . . . 80

6.2 An example of a chromosome with 3 aircraft and 5 flights . . . . 85

6.3 An example of a chromosome with assigned flights for each aircraft stored as a linked list for program execution . . . 85

6.4 Chromosome with aircraft one below the other to explain GA mechanics . . . 86

6.5 Random flight assignment process for each aircraft in the population . . . 88

6.6 An example of aircraft shift mutation . . . 95

6.7 An example of aircraft exchange mutation . . . 96

6.8 An example of flight exchange mutation . . . 98

6.9 An example of populate open spaces operators . . . 99

6.10 Uniform crossover example . . . 101

6.11 Single point crossover example . . . 103

6.12 Double point crossover example . . . 105

6.13 Process flow for the genetic algorithm solver . . . 106

6.14 Process flow for creating a CPLEX LP file from flight data . . . 109

6.15 Process flow for converting results from the time-space model to an aircraft-time line . . . 112

6.16 Flight representation for aircraft 1 using the NLIP FAM . . . . 114

6.17 Flight representation for aircraft 1 using the MCNF FAM . . . . 115

8.1 An example of two solutions with the same objective function . 120 8.2 MCNF FAM for flights in Table 6.3 . . . 123

8.3 Maintenance for time-space fleet assignment model . . . 125

8.4 NLIP FAM with maintenance . . . 126

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B.1 Input data flight network . . . 152

D.1 Flight duration percentage summary . . . 154

J.1 Data set 1 solution results . . . 167

K.1 Data set 2 solution results . . . 168

L.1 Data set 3 solution results . . . 169

M.1 Data set 4 solution results . . . 170

N.1 Data set 5 solution results . . . 171

O.1 Data set 6 solution results . . . 172

P.1 Data set 7 solution results . . . 173

Q.1 Data set 8 solution results . . . 174

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List of Tables

Page

4.1 Decision variables and their effect on minimum ground time

o_l1,l2 for each element x_l1,po_l1,l2x_l2,p . . . 60 4.2 Decision variables and their effect on fl1,l2 and hl1,l2 values for

each element fl1,l2(1 − hl1,l2) . . . 62

6.1 Data set information . . . 78 6.2 Fleet types utilised by airline with aircraft allocation and

associated hourly flying costs . . . 79 6.3 Sample airline schedule . . . 83 6.4 Creation of the Γ matrix ensuring minimum ground time and

conservation of aircraft flow constraints for the flights in Table 6.3 . . . 84 6.5 An example of an adjacency list for 4 flights . . . 84 6.6 Demonstration of how flights can be left out in initial solution

for a single aircraft . . . 89 6.7 Genetic algorithm standard operators and when they are executed107 6.8 Aircraft allocation for each fleet type . . . 111 6.9 MCNF FAM and NLIP FAM results for the aircraft data in

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6.10 MCNF FAM and NLIP FAM results when using 5 Airbus A319 aircraft . . . 113 6.11 Common flights found for each aircraft in solutions for the

MCNF FAM and NLIP FAM . . . 113

7.1 Performance comparison: MCNF FAM vs NLIP FAM . . . 117

7.2 Performance comparison for fleet assignment and aircraft

routing: MCNF FAM vs NLIP FAM . . . 118

7.3 Assignment cost comparison: MCNF FAM vs NLIP FAM . . . . 118

8.1 Γ matrix ensuring minimum ground time and conservation of

aircraft flow constraints for the flights in Table 6.3 . . . 123 8.2 An example of an adjacency list for 4 flights . . . 124 8.3 Decision variables and their effect on b_l1,l2 and g_l1,l2 values for

each element b_l1,l2(1 − g_l1,l2) . . . 132

8.4 Flight details and copies for MCNF FAM and NLIP FAM in

Figure 8.5 . . . 134 8.5 Decision variables xl1,p,n1 and xl2,p,n2 and their effect on the

ol1,l2,n1,n2 values . . . 139

8.6 Decision variables x_l1,p,n1 and x_l2,p,n2 and their effect on

f_l1,l2,n1,n2 and h_l1,l2,n1,n2 values . . . 139 A.1 Data set sample . . . 151 C.1 Flight duration summary . . . 153 S.1 The time taken by GA solver to obtain first solution complying

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Chapter 1 Introduction

In airline passenger transportation, profitability is influenced by an airline’s planning mechanism and market access. Thus airlines engage in a complex process of airline schedule planning.

Aircraft seats are a perishable product for an airline. A larger quantity secures more sales but also incurs costs (Du & Pardalos 2013). It is therefore incumbent for any airline to ensure that they have the correct fleet for each market.

While financially rewarding, Du & Pardalos (2013) indicate that the nature of the airline business can be characterised by the following attributes:

• Severe competition among airlines. • Large operational scale and scope.

• Tight coupling of resources such as aircraft, crew, maintenance facilities and airports.

• Active interactions and dependencies for all involved components. • A dynamic environment that is prone to disruptions.

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• Complex policies, business rules and tight control by the aviation au-thorities.

• Complex scheduling of routes and tasks. • Real-time and mission-criticality of decisions.

Some of the above attributes can be solved through techniques of opera-tions research. Lohatepanont (2002) identifies the airline scheduling processes, which require operations research, as:

• Schedule design, which is responsible for the creation of an optimal sched-ule.

• Fleet assignment for assigning fleet types to flights.

• Aircraft rotation for the optimal routing of specific aircraft in the sched-ule interspersed by the maintenance of each aircraft.

• Crew scheduling, which pertains to optimally assigning airline crew to flights in order to minimise costs.

The focus of this dissertation is on the fleet assignment process. This is a process that airlines use to assign aircraft fleet types to flights in order to maximise revenue and minimise operating cost. By using fleet assignment models, major airlines have reported significant profits. The model by Abara (1989) resulted in a 1.4% improvement in operational cost margins at Ameri-can Airlines. Similarly, Rushmeier & Kontogiorgis (1997) reported an annual profit increase of at least $15 million at US Airways through the use of fleet assignment models.

The fleet assignment process is part of a multi-step sequential process in-volving schedule design, fleet assignment, aircraft rotation and crew scheduling. The quality of the fleet assignment plan can have a major impact on airline

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profitability. Particularly if the assignment of aircraft is able to anticipate air-craft rotation and crew scheduling requirements as well as conform to a flexible schedule. The sequential execution of this multi-step process may result in the following disadvantages:

• Executing fleet assignment after schedule design, may result in an airline missing out on connections which improve revenue and similarly reduce cost.

• An airline may incur unnecessary layover costs in which the crew has to sleep over at their destination due to unconsidered flying time require-ments in the schedule.

• Assigning fleet types to flights and then performing maintenance routing, may result in the connection time between flights being insufficient to ensure consistent maintenance of all aircraft after the required number of flying hours.

Therefore, the objective of this research is to develop an alternative fleet as-signment model and compare it with the time-space multi-commodity network model proposed by Hane et al. (1995). The proposed fleet assignment model is designed to potentially allow easy integration with other airline decision pro-cesses resulting in improved profitability. To achieve this objective, it is shown how the proposed model can integrate with maintenance scheduling and air-craft routing. It is also shown how this model can be changed to allow for departure time flexibility.

The proposed mathematical model provides a solution to a problem put forward at a conference by a company which designs flight scheduling software called Fujitsu. The brief indicated that a mathematical model was required which would provide the following two advantages:

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1. It needs to easily integrate fleet assignment and other airline decision processes such as aircraft routing and maintenance scheduling.

2. It needs to be easy to interpret and implement for commercial airlines. Fujitsu also made available schedule data from a French commercial airline to allow for testing. To maintain the anonymity of the airline in this dissertation, this airline shall be referred to as Airline A.

1.1 Dissertation outline

1. An overview of the airline planning process is conducted in Chapter 2 and the importance of the fleet assignment process is indicated. This overview is followed by a literature review of the fleet assignment process which is given in Chapter 3.

2. In Chapter 4, the proposed non-linear fleet assignment model is pre-sented.

3. The proposed model is solved using a meta-heuristic, the genetic algo-rithm (GA). Chapter 5 is used to give an overview of this algoalgo-rithm. The details of how this algorithm is used for the proposed mathemati-cal model is shown in Chapter 6. In Chapter 7, the performance with respect to time and cost of assignment between the proposed model and the multi-commodity model is shown.

4. A discussion of the test results is presented in Chapter 8. This is fol-lowed by a consideration of how the meta-heuristic can be optimised to generate solutions faster. Although not tested in this dissertation, the benefit of using the proposed model is discussed in Chapter 8 through the introduction of departure time flexibility. The changes required on the model to accommodate maintenance scheduling are also discussed.

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The fleet assignment experimentation is conducted on a flight schedule provided by Airline A. This schedule has 21 384 flights for a period of 5 months. Each flight in the data set has a departure airport, arrival airport, departure time and arrival time. A data set sample is provided in Appendix A, Table A.1.

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Chapter 2 Literature Review - Airline

Planning Process

In this chapter, we will provide an overview of the the airline planning process which can be summarised into:

• Network design, and • Operational planning

2.1 Network design

The focus of the network design stage in the airline industry is primarily to find the optimal network structure and optimal routes to carry the targeted passenger flow at the lowest total transportation cost. This objective was not a priority prior to the Airline Deregulation Act of 1978 as the routes of US airlines were controlled by the Civil Aeronautics Board (CAB). The CAB required that airlines needed to show that proposed services would benefit the public and would not adversely affect current competitors in the market. Therefore, using long point-to-point routes was the norm (Du & Pardalos 2013).

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Airline deregulation brought about significant changes to even the most basic of airline operations, including fares, services, quality and safety. Du & Pardalos (2013) noted that 18 months after deregulation, 106 000 city-pair authorisations were approved compared to 24 000 granted 18 months before deregulation resulting in an increase in active point-to-point routes. The graph below shows growth in the number of active point-to-point routes from 1980 to 2010 which have more than doubled from 6 000 to over 15 000:

Figure 2.1: The number of active point-to-point air services by year (Pearce 2013)

A significant development for the airline industry was the adoption of hub-and-spoke networks. With this type of network, an airline would have a central hub and multiple non-hub airports. Services would be offered between the hub airport and other non-hub airports. An example of such a network is shown in Figure 2.2 below where Denver is the hub for all the other airports in the network.

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Figure 2.2: An example of a typical hub-and-spoke network (Flynn 2016)

According to Du & Pardalos (2013), significant research has been conducted on hub-and-spoke networks. This research either assessed the advantages of such networks in terms of airline economics or focused on mathematical mod-els to identify optimal hub locations. This can be seen in research by McShan & Windle (1989) who measured competitiveness in the airline industry after deregulation. McShan & Windle (1989) concluded that hub-and-spoke net-works would likely result in frequent flights and therefore an improved service. A further conclusion from their research is that total distance travelled by airlines utilising a hub-and-spoke network is reduced. O’Kelly et al. (1996) showed that, for airlines using a hub-and-spoke network, costs increase at a decreasing rate as passenger flow increases. This is in contrast to point-to-point routes, which do not take advantage of economies of scale from added passen-gers at the hub airport. Bailey et al. (1985) concluded that the hub-and-spoke network allows airlines to have more frequent flights with larger aircraft and a higher percentage of seat occupancy. Morrison & Winston (1986) investigated the benefit for passengers who use airlines which utilise hub-and-spoke net-works. They concluded that passengers benefit from hub-and-spoke networks as they have lower fares and shorter travel times.

Other research on the airline network design focused on the number of stops. Jaillet et al. (1996) conducted work on policy classification and defined several classifications, namely, stop”, “two-stop” and “all-stop”. For

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“one-stop” routes, an aircraft would fly passengers from one airport directly to another. A “two-stop” route happens when an airline provides an additional connection. An “all-stop” route which is an assumed policy in a monopolistic market happens when an airline caters for the maximum number of stops. According to Du & Pardalos (2013), most airlines in the US provide at most two stops for some routes. The rationale is that it is more profitable for airlines and does not make air travel too undesirable for passengers. The conclusion from Jaillet et al. (1996) is that a cost effective network design appears to be hub-and-spoke structured. They further recommended that airlines should adopt a “one-stop” policy. This is to provide for social factors such as passenger arrival time and to reduce the length of trips.

2.2 Operational planning

Operational planning involves the sequential steps shown in Figure 2.3. Strate-gic decisions require a few years to be executed while other tactical decisions are taken on a daily basis. The long-term decisions involve the mix of aircraft utilised which is decided on during the Fleet Planning stage. This is followed by Schedule Planning in which airlines determine the routes they will fly and the development of a schedule which encompasses schedule design, fleet as-signment and aircraft rotation. Near the time of flying, Revenue Management which involves prices of seats and seat inventory control is implemented. Rev-enue Management is conducted simultaneously with crew scheduling for each flight and scheduling of airport resources. All these steps which are shown in the figure below are explained in the sections that follow:

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Time Horizon T yp e of decision LONG SHOR T STRA TEGIC T A CTICAL Fleet Planning Schedule Planning - Route development - Schedule development * Schedule design * Fleet Assignment * Aircraft rotation Revenue Management - Pricing - Seat inventory control Crew Scheduling Airport Resource Planning

Figure 2.3: Overview of airline planning process (Lohatepanont 2002)

2.2.1 Fleet planning

Fleet planning is a process whereby an airline decides how many and what type of fleet types it should acquire or lease. This is a strategic decision which may involve a huge capital investment. A major re-vamp of a fleet or major changes in the fleet are done infrequently by airlines as each decision has a long-lasting impact on profitability. According to Belobaba (2009), there are two major approaches to fleet planning, namely:

• “Top-Down” approach, and • “Bottom-Up” approach.

The “Top-Down” approach involves a system wide financial analysis and normally uses market information to estimate demand, revenue and costs. The “Bottom-Up” approach however simulates the “to-be” airline operations. The

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success of the “Bottom-Up” approach is dependent on the richness of data especially for detailed forecasts and future operational scenarios.

In the recent past, a combination of high fuel costs as well as competition has led to the introduction of low-cost carriers. These “no frills” airlines have fewer fleet types, plan for high aircraft utilisation and target specific point-to point routes. The level of competition is exacerbated by the manufacturing of long-range, minimum-stops passenger aircraft.

Once the fleet has been selected by an airline, there is a long-term commit-ment. Thus, all other planning processes will take the fleet family as given. Decisions made at this level significantly affect down-stream decision processes.

2.2.2 Schedule planning

The schedule planning step begins at least 12 months before the schedule goes into operation. Airlines use the current schedule as a base and make modifications to account for market changes.

The three major activities of schedule planning are:

• Schedule design, • Fleet assignment, and

• Aircraft routing with maintenance scheduling.

Schedule design

To increase revenue, airlines have to optimise the use of given resources. The airline schedule is drafted a few months before it is executed. This complex stage can be broken down into two steps, namely:

• Frequency planning, and • Timetable development.

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For frequency planning, it is the duty of planners to determine the appro-priate service frequency in a market. Once flying frequency is determined, a timetable is developed. According to Du & Pardalos (2013), the factors that must be considered to draft an effective schedule are:

• Passenger demand at each airport in conjunction with the level of com-petition.

• Route features such as distances, operational restrictions, aircraft char-acteristics, flying speed and fuel cost.

Since profit is dependent on market share, maximisation of market share with limited aircraft capacity is the goal of an airline. Teodorovi (1988) showed that market share on routes with a large number of competitors is determined largely by flight frequency. Another important consideration is passenger seg-mentation. If the market is a long-haul international destination, the airline might be able to only offer a limited number of daily flights. A market domi-nated by business travellers requires frequent flight availability and convenience to perform connections at hub airports. Teodorovi´c & Krcmar-Nozi´c (1989) presented a multi-criteria model that aims to incorporate the major consider-ations for a good flight schedule in a competitive environment. These criteria focused on the following elements:

• Profit maximisation,

• Minimisation of passenger delays, and

• Maximisation of the number of passengers captured.

After the airline decides the number of flights it wants to offer in a market, the timing of those flights is determined. The main dependencies are market characteristics which include business, leisure or international travellers. Other

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dependencies are informed by schedule constraints such as airport constraints, personnel constraints (airport and crew) and market peak-period considera-tions (Lohatepanont 2002).

The time-line diagram in Figure 2.4 shows the domestic aircraft scheduling process at a major US airline. Between 5 and 10 years before departure, the airline engages in a fleet and route development process. The period between 1 year and immediately before departure can be summarised into the following activities:

• Schedule planning (365 days to 90 days before departure), which has an output of the desired schedule.

• Intermediate scheduling (90 days to 75 days before departure), where demand which affects frequency is revised.

• Current scheduling (75 days to 45 days before departure), which takes market factors into consideration. After this point the schedule is fixed and only aircraft could be changed due to unavailability and aircraft maintenance.

All other scheduling after this time considers other elements of the airline planning process which are crew scheduling, airport resource planning and marketing activities (Goodstein 1997). The cost of each flight is also deter-mined and profitability maximising strategies are implemented. Parking for each aircraft is also arranged and the schedule communicated. Below is the time-line for the airline with all the milestones mentioned above for schedule planning, intermediate planning, current scheduling as well as crew and airport resource planning:

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365 90 75 45 30 0 Time (days before departure)

Profit focus Feasibility focus Schedule planning Intermediate planning Current scheduling Crew scheduling Airport resource planning

Departure

Figure 2.4: The Domestic Airline Scheduling Process at a major US airline (Goodstein 1997)

Fleet assignment

During the fleet assignment activity, airlines assign available fleet types to every flight leg such that seating capacity on the aircraft closely matches the demand. A comprehensive literature study of the fleet assignment process is included in Chapter 3 of this dissertation.

Aircraft rotation and maintenance scheduling

Aircraft rotation and maintenance scheduling involves the compilation of a maintenance feasible routing of aircraft. Traditional fleet assignment models follow a sequential process in that flights are assigned to fleet types according to the number of available aircraft per fleet. Thereafter, the aircraft routing is determined for each aircraft in the fleet. Lohatepanont (2002) therefore high-lights that for many fleet assignment models including those by Hane et al. (1995), and Clarke et al. (1996), maintenance scheduling is approximated. In this approximation, airlines ensure a sufficient number of maintenance op-portunities are available for each fleet type. While this may ensure that on average, all fleet types are in maintenance every night, it does not guarantee that individual aircraft are treated equally (Lohatepanont 2002).

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Since airlines must meet the required standards for maintenance of air-craft, Sriram & Haghani (2003) indicate that there are normally three types of maintenance checks performed. These checks range from pilot inspections after each flight to type C checks during which each aircraft is rebuilt from scratch every few years. According to Sriram & Haghani (2003), type A checks are the shortest in duration and they normally take 4 hours. Sriram & Haghani (2003) further note that type A checks are performed every 40 to 65 flight hours and involve inspection of all major systems such as landing gear, engines and con-trol surfaces. Type B checks are performed every 300 to 600 flight hours and entail a thorough visual inspection plus the lubrication of all moving parts such as horizontal stabilisers. These type B checks normally require 15 hours for each aircraft. Sriram & Haghani (2003) indicate that the type C check is done on each aircraft once every 4 years and requires each aircraft being ser-viced not to be scheduled for a period of up to a month. According to Sriram & Haghani (2003), the principal requirement for optimising maintenance for airlines is in meeting type A checks and type B checks.

2.2.3 Revenue management

Given a schedule, the objective during the revenue management activity is to use optimization tools to maximise profitability. The fare levels offered for a flight are affected by market factors such as competitor fares. In order to be effective in a competitive environment, many airlines offer a wide variety of fares. Belobaba (1987) identifies two distinct but closely related components for revenue management, namely:

• Differential pricing, and • Seat inventory control.

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Pricing

Many airlines use differential pricing, that is, offering different fare products with different restrictions and services at different prices. According to Belob-aba (1987), this concept takes advantage of the customers’ willingness to pay. The result is that the same product is sold at different prices to customers, depending on the customers’ perception of value. The objective of differential pricing is to stimulate demand with low-fare offerings and to capture the will-ingness to pay of high-fare passengers. Thus an aspect of revenue management is to balance the number of discount and full-fare reservations accepted for a flight. The need to balance this aspect of pricing comes from the result that lower fares attract more passengers, thus creating greater load factors. How-ever, they also take away seats which could have been sold at a higher margin. Fare restrictions are used to prevent demand dilution from diversion, which happens when existing high-fare passengers opt to take advantage of low fare offerings.

Seat inventory control

With seat inventory control, airlines limit low-fare seats and protect high-fare seats for later booking passengers. According to The revenue enhancement po-tential of airline yield management systems (1992), several methods are utilised

by airlines to achieve this objective, namely:

• Overbooking: acceptance of bookings in excess of capacity.

• Fare class mix: limiting the availability of seats sold at various price levels on a flight leg.

• Itinerary control: discrimination of passengers depending on their itineraries. Most airlines manage seat allocation on a flight-by-flight basis. This is due to the complexity of the seat inventory control process. If we consider

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an airline with a hub-and-spoke network, every flight from the hub can have passengers going to any of the spoke airports. Every flight to the hub can have passengers from all of its spoke airports. In addition, every flight has several fare levels and this is over and above the issue of passenger demand which is not deterministic. Thus Du & Pardalos (2013) conclude that to build and solve a model optimising seat utilisation, which covers all the decisions for all the combinations of flights, and also fully address the issue of customer demand for each flight is impossible. Thus all seat inventory models have built-in assumptions which make modelling possible.

Strategic alliances

Strategic alliances are used to maximise profitability for an airline. They allow airlines to provide a service beyond the major city in another country. Accord-ing to Whalen (2007), this partnership can take on several forms dependAccord-ing on integration between airlines, namely:

• Code-sharing alliance, and • Antitrust immunised alliance.

Pierce & Doernhoefer (2011) indicate that the benefit for airlines and cus-tomers due to code-sharing are:

• A seamless booking experience as the marketing airline puts its ticket code on a connecting flight operated by another airline.

• A much more coordinated service in terms of flight times between mul-tiple airlines.

• Interline passengers benefit from lower fares behind and beyond interna-tional hub airports.

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• Feeder routes allow airlines to operate larger aircraft in city pairs which would otherwise not have the required demand.

2.2.4 Crew scheduling

In crew scheduling, an airline’s objective is to find a minimum cost assign-ment of flight crew (pilots and/or attendants) to flight legs. This process is subject to restrictions on qualified pilots to fly only a certain aircraft type, maximum time-away-from-base requirement and maximum flying time require-ment. Coupled with these restrictions are union contract considerations and provision of rest time for personnel. For most airlines, the crew expense is the second largest cost component, second only to the fuel expense, thus a small improvement in crew scheduling can lead to millions in savings. This has driven a lot of focus on finding optimal ways to schedule airline crews. Barnhart & Talluri (1997) break crew scheduling into the following elements:

• Crew-pairing problem, and • Crew assignment problem.

The objective during crew pairing is to find work schedules that cover each flight and also minimises total crew cost. A pairing is made up of duties segmented by rest periods where a duty is a consecutive number of flights flown on a single day which satisfies all work rules. Sometimes, a crew member need not be assigned to a connected sequence of flights. The disconnection is fixed by dead-heading where a flight crew member is repositioned by flying as a passenger. Barnhart & Talluri (1997) showed the advantages of dead-heading especially in long-haul crew pairing problems.

In crew assignment, a crew schedule is combined manually with rest pe-riods, training and vacations to create an extended plan for each individual. Lohatepanont (2002) classified the traditional methods for crew assignment as:

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• Rostering, and • Bid-line generation.

With rostering, which is mainly used in Europe, schedules are constructed for specific individuals with a subset of schedules selected so that each indi-vidual is assigned to a schedule. For bid-line generation, an airline assigns crew according to preferences from a bidding process with senior staff getting priority.

2.2.5 Airport resource planning

Airport resource planning is an operational process in which the airline al-locates gates for aircraft and schedules ground personnel. Gates would be allocated to arriving and departing aircraft. The allocation also ensures that all flight legs are covered, aircraft maintenance is conducted and passenger connections can be made within reasonable time Lohatepanont (2002).

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Chapter 3 Fleet Assignment

In the airline planning process from Chapter 2, it was shown that fleet assign-ment is one of four operational planning processes executed by airlines. The focus of this chapter is to delve into this process. This is done by provid-ing a historical overview of research relatprovid-ing to the fleet assignment process and indicating how disruptions affect fleet assignment planning. A synopsis of meta-heuristics used for solving the fleet assignment problem is also provided as the proposed solution will also use meta-heuristics. The time-space multi-commodity network fleet assignment model which is the basis for most fleet assignment models is also presented.

3.1 Fleet assignment overview

The earliest application of operations research techniques for airline scheduling was shown by Dantzig & Ferguson (1954). They considered fleet assignment for non-stop routes. These are pre-defined routes between multiple airports which equal the number of available aircraft. In their model, a case study with 69 aircraft of four fleet types, catering to a demand of 124 000 passengers was conducted. The solution approach involved assigning each aircraft to a route until all aircraft in the fleet are utilised or all passengers are served. After the system cost is determined, an iterative process of shifting aircraft between

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routes was adopted until there were no more opportunities for cost saving. Daskin & Panayotopoulos (1989) proposed an integer linear programming model that assigns aircraft to routes. Their formulation depends on a two-step process to identify feasible routes from the hub to multiple cities and back to the hub for a specific time-frame. This is followed by the assignment of the available aircraft to those routes while maximising profitability. They used a Lagrangian relaxation technique to obtain a solution.

Abara (1989) presented an integer programming model that uses underly-ing connection arcs as decision variables to assign fleet types to flights legs. This was applied to a daily schedule which ensured that the same number of aircraft of a specific fleet type were available every morning. Abara (1989) proposed a penalty on the objective function which is related to the shortage of originating and terminating fleet types at each airport. Because this model is dependent on available connections for each fleet type, the number of vari-ables could easily grow to an unmanageable size due to the number of possible connections. According to Sherali et al. (2006), another limitation is that dif-ferent flying times and turn around times (minimum ground service times) are not allowed with the model presented by Abara (1989). Abara (1989) counters the explosion of variables by specifying a limit on the number of connections. The model utilised by Abara (1989) resulted in a 1.4% reduction in cost at American Airlines.

In airline scheduling, profitability is enhanced by the number of passen-gers carried. In order to maximise profitability, Berge & Hopperstad (1993) developed an approach known as Demand Driven Dispatch (D3) that

identi-fies capacity reassignments as a departure date approaches. This model takes advantage of a demand forecast near departure in order to effect aircraft and crew swaps without affecting operations. For their data sets, this lead to cost savings of between 1% and 5%. Other research on this topic includes the work

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done by Sherali and Zhu (2008) who model stochastic passenger demand. Sher-ali et al. (2013a) also presented a model which uses optional flight legs in order to maximise profitability. For this model, the flight leg to be flown is selected while addressing recapture of customers and fleet assignment.

Hane et al. (1995) modelled the fleet assignment problem as a time-space multi-commodity network flow fleet assignment model (MCNF FAM). In this model, fleet types are assigned to flight legs in the network using the available fleet of aircraft. In comparison with the “connection network” model presented by Abara (1989), this model focused on representing flight legs in a time-space network. The number of decision variables was reduced as the number of flight legs is normally far less compared to the number of connections. However, Rushmeier & Kontogiorgis (1997) pointed out that the MCNF FAM is not able to distinguish between aircraft that are on the ground. Therefore, this model has limited usability especially for aircraft routing.

Hane et al. (1995) also introduced pre-processing techniques which reduce the network size. The first pre-processing technique takes advantage of the observation that as long as the correct connections are represented, consecu-tive arrivals and subsequent consecuconsecu-tive departures can share a single node. In the experiments conducted by Hane et al. (1995), this technique, called node aggregation, reduced the number of rows by a factor of 3 to 6 and the number of columns by a factor of 1 to 3. The second pre-processing technique assumes that for hub-and-spoke networks, there are times, especially at the spoke air-port with sparse flight activity where there are no aircraft on the ground at that airport. These ‘zero-valued’ flows with no aircraft on the ground are nominated for deletion from the network. This simplification results in the creation of islands so that some nodes would have the same number of arrival and departure flights. The third pre-processing technique eliminates missed connections where two flights that are assigned to the same fleet type cannot

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connect due to longer turn around times for that fleet type. Hane et al. (1995) utilised these pre-processing techniques on a specific problem instance. They reported a reduction in the problem size from 48 982 rows and 66 942 columns to 7 703 rows and 20 464 columns.

Rushmeier & Kontogiorgis (1997) used the same model as Abara (1989) with added pre-processing to create connection complexes with an equal num-ber of arrival and departure legs representing possible connections. They demonstrated that these complexes reduce the solution time substantially with-out an adverse effect on accuracy or profit. Rushmeier & Kontogiorgis (1997) reported an annual saving of at least $15 million at US Airways.

The MCNF FAM optimises profitability for each flight leg, however, many customers fly multiple legs. The flight legs in this model are independent, thus many airlines lose revenue due to spilled passengers. These are passengers who cannot be accommodated in subsequent flights due to capacity constraints. Jacobs et al. (1999) derived an iterative method for solving the fleet assignment model that enhances the spill estimation process which is static in the model presented by Hane et al. (1995). The algorithm proposed by Jacobs et al. (1999) begins by solving a special relaxation of the fleet assignment model on an instance of estimated passenger flow. The results are then used to revise passenger flow in the network. The algorithm keeps on iterating until all constraints are satisfied and an integer solution is obtained.

In order to improve profitability, Rexing et al. (2000) took an approach that considered integrating fleet assignment with other airline decision processes. They presented an expanded fleet assignment model, based on the model by Hane et al. (1995). Their expanded fleet assignment model allows for re-timings of nodes within small time windows. The result is the integration of flight scheduling and fleet assignment, which is restricted by the time window between legs. For this model, the set of departure and arrival times is specified

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for a particular time window. Each possible departure node is connected to an aligned arrival node by copies of flight legs. Only one of these copies needs to be flown. Rexing et al. (2000) reported that this extra departure time flexibility resulted in a cost saving of over $67 000 per day for 10 minute time windows at a major US airline.

According to Belanger, Desaulniers, Soumis & Desrosiers (2006), the model proposed by Rexing et al. (2000) at times generates slightly inaccurate prof-itability due to departure times of flights with the same origin-destination (O-D) pairs being too close. The result is an overestimation in passenger demand and overstated pricing as those flights effectively compete for customers. Be-langer, Desaulniers, Soumis & Desrosiers (2006) thus proposed a periodic fleet assignment model with time windows and used branch-and-bound as well as column generation to obtain a solution. In this model, penalties are introduced for short spacings for flights with the same O-D pairs.

For increased profitability, Barnhart et al. (2002) showed that there is an increased improvement in profitability from including network effects which approximate the number of spilled passengers in the fleeting process. They also concluded that there is an even bigger improvement by including recapture where passengers are redirected from their desired itinerary to an alternate itinerary. They observed that with the fleet assignment model proposed by Hane et al. (1995), network effects cannot be determined and passengers on multi-leg itineraries can be spilled from one flight leg but not the others. Their extended version of the fleet assignment model is called the Itinerary-Based Fleet Assignment Model (IFAM). It was shown that the IFAM model is better able to make fleeting decisions that allow high revenue passengers to be carried and low revenue passengers to be spilled. It is worth noting that opportunities for successful recapturing diminish as capacity on alternate itineraries becomes more fully assigned.

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3.2 Fleet assignment during disruptions

Flight delays not only increase operational costs of the delayed flight, but sometimes affect downstream flights. They also inconvenience passengers and the airline’s credibility is damaged. The most common factor causing delays is inclement weather and since airlines cannot control the weather, they have to create strategies to minimise the impact. According to Du & Pardalos (2013), there are two types of delays, which are ground delays and airborne delays. Ground delays affect airlines before take-off and airborne delays are caused by a delay in landing an aircraft due to unforeseen circumstances at an airport. When there are delays, airlines are faced with a decision of cancelling several flight legs.

One solution to manage disruptions is swapping aircraft at specific air-ports to cater for changed load factors and disruptions due to mechanical faults. Clarke & Laporte (1997) however indicates that swapping aircraft dur-ing disruptions is problematic as it affects the crew, as the specifications of the aircraft being swapped may not be the same. A different strategy for swap-ping aircraft has been suggested by Ageeva & Clarke (2000) where routes are overlapped within an aircraft rotation at hubs. This gives an opportunity to swap aircraft should there be a disruption affecting a specific aircraft. In or-der to minimise the swapping of the crew when aircraft are swapped, Smith & Johnson (2006) showed that aircraft swaps can be effectively done by imposing station purity. For effective station purity, the number of fleet types serving a given station should not exceed a specified limit, and the fleet types in the model are defined as crew-compatible families.

Rosenberger et al. (2002) described a way in which airlines can recover operations by cancelling a minimum number of flights during disruptions. This is done by the creation of short-cycles within fleet assignment for each aircraft path. With this string-based fleet assignment model, airlines are able to cancel

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a minimum number of flights while an aircraft remains at an airport only to depart from that airport later. This model is better executed in hub-and-spoke networks. The other possibility, according to Rosenberger et al. (2004), is for airlines to ferry the aircraft to the next airport without passengers. This is normally the last option due to cost implications.

Rosenberger et al. (2004) provided another strategy for airlines to recover operations and minimise the impact to downstream flights by isolating hubs. This is effected by adding a constraint within fleet assignment such that flights from one hub to another are minimised so that a disruption at a single hub does not affect other connected hubs.

3.3 Fleet assignment meta-heuristics

Meta-heuristics have been widely used for airline schedule planning. For fleet assignment, the genetic algorithm has been used extensively in optimising a given schedule. This can be observed in the contribution by Lee et al. (2007) where a given schedule is optimised in order to minimise the impact of disruptions from delays. This is done by re-timing of departure times for all flights in order to minimise losses from disruptions. A similar objective is achieved by Burke et al. (2010) who pro-actively monitor disruptions for a given airline and use the genetic algorithm to create departure time flexibility by re-timing flights. The recovery strategies used in both models involved swapping aircraft, cancelling flights and accepting delays. For both models, the genetic algorithm is applied to an already defined schedule which has been constructed using the MCNF FAM proposed by Hane et al. (1995).

Other uses of the genetic algorithm for airline scheduling are observed in contributions by Christou et al. (1999) and Ozdemir & Mohan (2001). For both contributions, genetic algorithms are used to schedule flight crews.

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3.4 Integrated fleet assignment models

The latest work on airline fleet assignment is focused mainly on the integra-tion of fleet assignment with other airline scheduling decisions. A notable integrated model is provided by Barnhart et al. (1998). In this model, “flight strings” which are the flights in a schedule are assigned to specific aircraft. The result is the creation of a routing for each aircraft which minimises cost. This mathematical model is solved using a two-step process of identifying all po-tential routes and thereafter assigning “strings” of routes to each aircraft. The model used a “depth-first-best-bound-depth-first” node choice rule to find an optimal solution. This rule ensured that the next node chosen is one with the best bound with a limitation of 1000 searched nodes set. This model therefore combined both fleet assignment as well as aircraft routing even though 90% of the solution time was spent in string generation for some cases. It was shown that for 190 flights, more than 500 million strings were generated. The solution times of up to 10 hours were observed, with the best solution time of 5 hours and 27 minutes for a tolerance of 1.00% compared to the MCNF FAM. This was obtained for 1 124 flights visiting 40 cities with 9 fleet types. The benefit of such a model was shown to be improved aircraft utilisation, the determination of aircraft rotation and the augmenting of maintenance to “strings” of flights so that all aircraft are adequately maintained.

Other integrated models are models by Rexing et al. (2000) which has been explained earlier. Sherali et al. (2013b) performed fleet assignment while con-sidering flight re-timing and the use of optional legs for flight flexibility. Pita et al. (2012) considered integrated flight scheduling and fleet assignment under airport congestion. Liang & Chaovalitwongse (2012) performed fleet assign-ment with maintenance routing. The bar graph in Figure 3.1 is created from the papers used to perform a literature study for this dissertation. It shows a summary of research for airline scheduling decisions. As can be seen, the

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number of publications focusing on integrated models has increased signifi-cantly. Fleet assignment and aircraft routing have also remained a focus for cost saving in airline scheduling:

Figure 3.1: Research published for airline scheduling decisions

3.5 Time-space multi-commodity network flow

model

According to Lohatepanont (2002), the basis for several fleet assignment mod-els used in industry is the MCNF FAM proposed by Hane et al. (1995). The objective of this model is to assign aircraft types to flight legs based on a fixed schedule while maximising revenue or minimising cost. Figure 3.2 shows a graphical representation of the model with diagonal lines between airports in-dicating the flights. Ground arcs are represented by the dashed lines and the points where each arc ends and others start are the nodes:

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A B C 08:00 09:00 10:00 11:00 ground arcs flight arcs Time (hrs) Airp ort

Figure 3.2: A representation of a time-space multi-commodity network flow fleet assignment model (MCNF FAM)

The MCNF FAM is defined as follows (Lohatepanont 2002):

Given a flight schedule with fixed departure times and cost (fleet

and flight specific operating and spill costs), find the minimum cost

assignment of aircraft types to flights, such that (1) each flight is

covered exactly once by a fleet type, (2) flow of aircraft by type is

conserved at each node, and (3) only the available number of aircraft

of each type are used.

While fleet assignment can also be analysed using maximisation of profit, minimising cost provides for an easier model which does not take into consider-ation fill factors and competition. Therefore, minimisconsider-ation is also adopted for this paper so that only the cost of each flight and not profitability is considered.

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3.5.1 Notation

Sets

E : The set of airports indexed by e.

L : The set of flight legs in the flight schedule indexed by l.

K : The set of different fleet types indexed by k.

T : The set of all event times (departure or arrival with minimum ground service time) at all airports, indexed by t. The event at time t occurs before the event at time t + 1.

N : The set of nodes in the time-line network indexed by k,e,t.

CL(k) : The set of flight legs flown by fleet type k during a nominated time defined as the count time.

I(k,e,t) : The set of inbound flight legs to node (k,e,t).

O(k,e,t) : The set of outbound flight legs to node (k,e,t).

Decision variables

y_(k,e,t+₎ : The number of aircraft of fleet type k ∈ K that are on the ground at airport e ∈ E immediately after time t ∈ T .

y_(k,e,t−₎ : The number of aircraft of fleet type k ∈ K that are on the ground at airport e ∈ E immediately before time t ∈ T .

y_(k,e,tm₎ : The number of aircraft of fleet type k ∈ K aircraft that are on the ground at airport e ∈ E during the nominated count time tm.

x_k,l ₌

  

1 If flight leg l ∈ L is assigned to fleet type k ∈ K

0 Otherwise

Parameters

qk : The number of aircraft of fleet type k, ∀k ∈ K.

c_k,l : The assignment cost when fleet type k ∈ K is assigned to flight leg l ∈ L.

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3.5.2 Mathematical model

The mathematical model is formulated as follows:

M inX l∈L X k∈K ck,lxk,l (3.1) Subject to: X k∈K xk,l= 1, ∀l ∈ L (3.2) y_(k,e,t−₎+ X l∈I(k,e,t) x_k,l− y_(k,e,t+₎− X l∈O(k,e,t) x_k,l= 0, ∀(k, e, t) (3.3) X e∈E y_(k,e,tm₎+ X l∈CL(k) xk,l≤ qk, ∀k ∈ K (3.4) xk,l∈ {0, 1}, ∀k ∈ K, ∀l ∈ L (3.5)

y_(k,e,t+₎, y_(k,e,t−₎, y_(k,e,tm₎≥ 0, ∀(k, e, t) (3.6)

Constraints (3.2) are cover constraints and they ensure that each flight is covered once by a fleet type. This attribute is made certain by equating the summation of each flight assignment for each fleet type to 1 and this is repeated for each leg. Therefore, only a single fleet type assignment can be performed for each flight leg. Constraint (3.3) conserves aircraft flow. For this constraint, the number of fleet types on the ground as well as fleet type arrivals immediately before each node are added. The number of fleet types on the ground as well as fleet type departures after the said node are also added. Balance is maintained for each node by calculating the difference between the summation of fleet type arrivals and ground fleet types before the node and the summation of fleet type departures with fleet types on the ground after the node. Constraints (3.4) are count constraints and ensure that only the available number of each aircraft

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for each fleet type are used in the assignment.

The objective function in equation (3.1) is made up of operating costs which include the cost of fuel, gate rental as well as airport costs. These costs are calculated for each flight and fleet type combination. For more advanced fleet assignment models, passengers who cannot be accommodated due to capacity constraints are spilled. A cost is estimated based on the number of spilled pas-sengers. Furthermore, an estimation can be determined for spilled passengers who are recaptured and flown on itineraries other than their desired itineraries Barnhart et al. (2002).

According to Lohatepanont (2002), the characteristics of this model are:

• The optimal assignment of fleet types to the available flights.

• Flight leg independence: The choice of one flight leg is not influenced by another.

• Equal fare allocation: because the MCNF FAM assumes flight leg inde-pendence, fare allocation for passengers flying multiple legs is the same as for passengers flying a single flight leg. No optimisation scheme is used to determine differences.

• Spill estimation: the way to determine spilled revenue is deterministic and it uses unconstrained demand to determine if all passengers have been carried.

• Individual aircraft utilisation is not accounted for as fleet types are as-signed and the route per aircraft is not established. Only the overall aircraft utilisation is accounted for.

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3.6 Conclusion

The MCNF FAM proposed by Hane et al. (1995) has led to many advances in fleet assignment modelling. These advances include:

(a) The pre-processing steps used by Hane et al. (1995) to reduce the number of variables.

(b) The introduction of flight copies to improve the number of potential flight connections proposed by Rexing et al. (2000).

(c) An extension of the flight copies model from Rexing et al. (2000) by Be-langer, Desaulniers, Soumis & Desrosiers (2006). They introduced penal-ties for short spacings for flights with the same O-D pairs in order to minimise impact of overestimated passenger demand.

(d) An extension of the MCNF FAM from Hane et al. (1995) by Jacobs et al. (1999) to improve the spill estimation of customers.

(e) An extension of the MCNF FAM from Hane et al. (1995) by Barnhart et al. (2002) who modelled the spill and recapture of customers.

The model by Hane et al. (1995) assigns aircraft fleet types to flights. Therefore, the route taken by each aircraft is not modelled. This extension was introduced by Barnhart et al. (1998) who used “flight strings” to model both the fleet assignment as well as aircraft routing. The “flight strings” model resulted in the simultaneous solution of fleet assignment and aircraft rotation which includes maintenance.

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Chapter 4 Proposed Non-linear Integer

Programming Fleet Assignment

Model

As shown in Chapter 3, the MCNF FAM proposed by Hane et al. (1995) (Chapter 3, Section 3.5) can be used or integrated with more elements or decisions of airline scheduling. In this chapter, we propose a non-linear integer programming fleet assignment model (NLIP FAM). An explanation as well as validation of this model is presented and its characteristics are provided.

4.1 Proposed fleet assignment model

Figure 4.1 shows a representative example of the proposed fleet assignment model. In this example, there are two aircraft, an Airbus A330 (aircraft 1) and an Airbus A320 (aircraft 2). According to Figure 4.1, the A330 aircraft will fly “flight 1” departing from “airport B” and arriving at “airport C”. It thereafter flies “flight 2” from “airport C” to “airport B”. The length of each flight represents the duration of the flight irrespective of the aircraft used. The time between “flight 1” and “flight 2” is denoted as the minimum ground time.

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This is the minimum time that each aircraft has on the ground for airline personnel to prepare the aircraft before departing on the next flight and it is the same for all aircraft. Similarly, the Airbus A320 aircraft departs from “airport A” at 08:00 flying “flight 3” and arrives at “airport B” at 09:00. This flight is followed by “flight 4” from “airport B” to “airport A”. Thereafter, “flight 5” from “airport A” to “airport C” is flown.

As can be seen from the example in Figure 4.1, unlike the MCNF FAM, the NLIP FAM provides a schedule for each aircraft and not for each fleet type. The proposed NLIP FAM thus has to be understood from the perspective that all aircraft belonging to an airline are part of its fleet. It is in this context that this model is also a fleet assignment model. This is also the reason why this model not only provides fleet assignment, but simultaneously solves aircraft routing. The NLIP FAM may be represented by an aircraft-time graph as seen in the example in Figure 4.1 below. This is different from the time-space representation of the MCNF FAM shown in Figure 3.2:

(2)A320 (1)A330 08:00 09:00 10:00 11:00 Time (hrs) Aircraft A 3 B B 4 A A 5 C B 1 C C 2 B

Departure airport for flight 1 represented by parameter d1

or more generally dlwhere

l is the flight leg.

Arrival airport for flight 2 represented by notation a2

or more generally alwhere

Flight 1 and flight 2 adhere to the minimum ground time (gtmin₎

and conservation of aircraft flow at airport C.

Departure time of flight 3 represented by parameter s3

or more generally slwhere

Arrival time of flight 4 represented by parameter t4

or more generally tlwhere

Figure 4.1: A graphical depiction of the proposed fleet assignment model with parameter descriptors

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The proposed mathematical model can be defined as follows:

Given a flight schedule with fixed departure times and cost per flight,

find the minimum cost of flights assigned to aircraft, such that (1)

each flight is covered exactly once by an aircraft, (2) aircraft depart

from the last airport which they landed on and (3) minimum ground

time between flights is maintained for each aircraft

4.1.1 Notation

Sets

P : The set of all aircraft indexed by p. |P | is the number of aircraft in P . For the example in Figure 4.1, |P | = 2. Dissimilar to the MCNF FAM which assigns flights to fleet types, the NLIP FAM assigns the flights to the actual aircraft and therefore an index for each fleet type is not required.

L : The set of all flight legs in a schedule indexed by l. This set is sim-ilar to that used for the MCNF FAM. |L| is the number of flight legs in L. For the example in Figure 4.1, |L| = 5.

Parameters

d_l : Departure airport of flight leg l ∈ L. al : Arrival airport of flight leg l ∈ L.

tl : Arrival time of flight leg l ∈ L.

sl : Departure time of flight leg l ∈ L.

gtmin : Required minimum ground time between any pair of flights flown by the same aircraft p ∈ P . This time allows for passengers to get off and flight crew to prepare an aircraft for the next flight.

λ : An all-one vector such that λ ∈ R|P |. τ _{: An all-one vector such that τ ∈ R}|L|. η : An all-one vector such that η ∈ R|L|2.

cl,p : The cost of flight l ∈ L flown by aircraft p ∈ P . C = (cl,p), ∀l ∈ L,

∀p ∈ P , and C ∈ R(|L| × |P |).

o_l1,l2 : The parameter that ensures minimum ground time between flights.

O = (o_l1,l2_{), ∀l1, l2 ∈ L and O ∈ R}(|L| × |L|).

fl1,l2 : The parameter that ensures conservation of aircraft flow between

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Flight cost coefficients

For each aircraft p ∈ P and flight l ∈ L, the matrix C ∈ R(|L| × |P |) stores each flight and aircraft cost element c_l,p∈ C. Here, l = 1 , 2 , 3 , · · · , |L| and p = 1 , 2 , 3 , · · · , |P | and the matrix

C =             c1,1 c1,2 · · · c1,|P | c2,1 c2,2 · · · c2,|P | .. . ... . .. ... c_|L|,1 c_|L|,2 · · · c_{|L|,|P |}             .

We now construct the vector c by vectorising the columns of C. Hence, vec(C) = c =

c1,1 · · · c|L|,1 c1,2· · · c|L|,2 · · · c1,|P | · · · c|L|,|P |

T

, where c ∈ Rr, for r = |L| × |P |. When |L| = 3 and |P | = 2, c =

c1,1 c2,1 c3,1 c1,2 c2,2 c3,2

T

.

Minimum ground time parameter

The parameter ol1,l2 is used to check if a pair of flights l1, l2 ∈ L comply with

the minimum required ground time when flown by any aircraft p ∈ P . If l1 = l2, ol1,l2 refers to the same flight and its value is made equal to 0. The values of

this parameter are shown as:

ol1,l2=          0 If (s_l2− tl1≥ gtmin) or (l1 = l2), 1 Otherwise. (4.1)

The matrix O ∈ R(|L| × |L|), is shown for all flights l1, l2 ∈ L. Each element ol1,l2∈ O indicates whether flights l1 and l2 comply with the minimum ground

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O =             o1,1 o1,2 · · · o1,|L| o2,1 o2,2 · · · o2,|L| .. . ... . .. ... o_|L|,1 o_|L|,2 · · · o_|L|,|L|             .

Therefore, each element of matrix O will either be 0 or 1 based on the calcu-lated parameter in equation (4.1). The diagonal elements of O are zeros.

Conservation of aircraft flow parameter

The parameter fl1,l2is used to calculate conservation of aircraft flow from flight

leg l1 ∈ L to flight leg l2 ∈ L. An IF-THEN statement showing the values of this parameter is shown in equation (4.2). In cases where l1 = l2, fl1,l2 refers

to the same flight. For this case, the value of f_l1,l2 is equal to 0. Hence,

fl1,l2=          0 If (d_l2= a_l1) or (l1 = l2), 1 Otherwise. (4.2)

The matrix F ∈ R(|L| × |L|) shows each element f_l1,l2∈ F for all combinations of flights l1, l2 ∈ L. The definition for “conservation of aircraft flow” from the MCNF FAM meant that for each node, the summation of the number of flights arriving at a node with flights on the ground before that node needed to equal the summation of the number of flights departing from the same node with flights on the ground after that node. Since there are no such nodes in the NLIP FAM, “conservation of aircraft flow” means that for any pair of consecutive flights, the arrival airport of the flight flown first is the same as the departure airport of the next flight. This result is ensured by the use of the parameter in equation (4.2) for any pair of flights l1, l2 ∈ L. A vector f is formed from the columns of matrix F such that f ∈ R|L|2. Therefore,

Non-linear integer programming fleet assignment model